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222 IX ANALYTIC FUNCTIONS

hence f /(*) dx = 0 by Cauchy's theorem, and this proves our assertion.
We can therefore define g(z) as the value of J/W dx for any road y in A
of origin a and extremity z, and by (9.7.2), g is defined in A. Now for any
z₀ e A, there is an open ball B c A of center z₀ in which f(z) is equal to a
convergent power series in z - z₀; by (9.3.7) there is therefore a primitive h
of/in B which is analytic, and such that h(z₀) = g(z₀); hence we have for
zeB

h(z) - ft(*₀) = f /(z₀ + *(* - z₀))(z - z₀) dr.
Jo

But the right-hand side is by definition f f(x) dx, where a is the road
I ^ Z_Q + /(z - _ZQ) defined in [0,1]; as that road is in B c: A, we have
g(z) - #(z₀) = f f(x) dx by definition of g, and therefore g(z) = h(z) in B.

J<r

Q.E.D.

8. INDEX OF A POINT WITH RESPECT TO A CIRCUIT

(9.8.1) Any path y defined in an interval I = [a, b] and such that y(I) is
contained in the unit circle U ={ze C | |z| = 1}, has the form t-^e^l^^(t\
where \jt is a continuous mapping of I into R; if y is a road, if/ is a primitive
of a regulated function.

As y is uniformly continuous in I, there is an increasing sequence of
points t_k (0 < k ^ p) in I such that t₀ = a, t_p — b, and that the oscillation
(Section 3.14) of y in each of the intervals l_k = [t_k, t_k+l] (0 < k ^p — 1)
be < 1. This implies that y(I_k) ^ U; if 9_k e R is such that e^idk <£ y(l_k) (9.5.7),
then x ~+ e^l(x+6k) is a homeomorphism of the interval ]0,2n[ on the complement
of e^ldk in U (9.5.7). If <p_k is the inverse homeomorphism, we can therefore
write, for t e I_k, y(t) = e^iM\ where \j/_k(t) = (p_k(y(t)) +0_fc is continuous in
l_k. By (9.5.5), we have ^+1(^+1) = W^+i) + 2n_kn with n_k an integer
(0 < k </? — 2). Define now \j/ in I in the following way: \//(t) = \l/₀(t) for
t e I₀; by induction on k, we put \l/(f) = \l/_k(t) + ^(^) - *A*(^) f or ^ < / < r_fc+_x.
By induction on k, it is immediately seen that ijs(t_k) - 1/^(4) is an integral
multiple of 2n for Q^k^p- 1; therefore 7(0 = e^w) for re I, and \j/
is obviously continuous in I. Moreover, if y(t) = a(r) + fj5(0, we have
a(*) = cos \l/(ty, ft(t) = sin \l/(t), and one of the numbers cos \l/(t), sin if/(t)
is not 0; from (9.5.4), and (8.2.3) applied to one of the functions cos*,
sin * at a point where it has a derivative ^ 0, we deduce that if y has a
derivative at a point t, so has \j/₉ and i\l/'(t) = /(OMO> which ends our proof.



	222 IX ANALYTIC FUNCTIONS hence f /() dx = 0 by Cauchy's theorem, and this proves our assertion. We can therefore define g(z)* as the value of J/W dx for any road y in A of origin a and extremity z, and by (9.7.2), g is defined in A. Now for any z₀ e A, there is an open ball B c A of center z₀ in which f(z) is equal to a convergent power series in z - z₀; by (9.3.7) there is therefore a primitive h of/in B which is analytic, and such that h(z₀) = g(z₀); hence we have for zeB h(z) - ft(₀) = f /(z₀ + (* - z₀))(z - z₀) dr. Jo But the right-hand side is by definition f f(x) dx, where a is the road I ^ Z_Q + /(z - _ZQ) defined in [0,1]; as that road is in B c: A, we have g(z) - #(z₀) = f f(x) dx by definition of g, and therefore g(z) = h(z) in B. J<r Q.E.D.

	8. INDEX OF A POINT WITH RESPECT TO A CIRCUIT (9.8.1) Any path y defined in an interval I = [a, b] and such that y(I) is contained in the unit circle U ={ze C \| \|z\| = 1}, has the form t-^e^l^^(t\ where \jt is a continuous mapping of I into R; if y is a road, if/ is a primitive of a regulated function. As y is uniformly continuous in I, there is an increasing sequence of points t_k (0 < k ^ p) in I such that t₀ = a, t_p — b, and that the oscillation (Section 3.14) of y in each of the intervals l_k = [t_k, t_k+l] (0 < k ^p — 1) be < 1. This implies that y(I_k) ^ U; if 9_k e R is such that e^idk <£ y(l_k) (9.5.7), then x ~+ e^l(x+6k) is a homeomorphism of the interval ]0,2n[ on the complement of e^ldk in U (9.5.7). If <p_k is the inverse homeomorphism, we can therefore write, for t e I_k, y(t) = e^iM\ where \j/_k(t) = (p_k(y(t)) +0_fc is continuous in l_k. By (9.5.5), we have ^+1(^+1) = W^+i) + 2n_kn with n_k an integer (0 < k </? — 2). Define now \j/ in I in the following way: \//(t) = \l/₀(t) for t e I₀; by induction on k, we put \l/(f) = \l/_k(t) + ^(^) - A(^) f or ^ < / < r_fc+_x. By induction on k, it is immediately seen that ijs(t_k) - 1/^(4) is an integral multiple of 2n for Q^k^p- 1; therefore 7(0 = e^w) for re I, and \j/ is obviously continuous in I. Moreover, if y(t) = a(r) + fj5(0, we have a() = cos \l/(ty, ft(t) =* sin \l/(t), and one of the numbers cos \l/(t), sin if/(t) is not 0; from (9.5.4), and (8.2.3) applied to one of the functions cos, sin at a point where it has a derivative ^ 0, we deduce that if y has a derivative at a point t, so has \j/₉ and i\l/'(t) = /(OMO> which ends our proof.